Nanovortex-driven all-dielectric optical diffusion boosting and

sorting concept for lab-on-a-chip platforms

Adrià Canós Valero1, Denis Kislov1, Egor A. Gurvitz1, Hadi K. Shamkhi1, Dmitrii Redka2,

Sergey Yankin3, Pavel Zemánek4 and Alexander S. Shalin1

1ITMO University, Kronverksky prospect 49, 197101, St. Petersburg, Russia

2Electrotechnical University “LETI” (ETU) 5 Prof. Popova Street, 197376, Saint Petersburg,

Russia

3 LLC COMSOL, Bolshaya Sadovaya St. 10, 123001, Moscow, Russia

4Czech Academy of Sciences, Institute of Scientific Instruments, Královopolská 147, 612 64

Brno, Czech Republic

Corresponding author: adria.canos@optomech.ifmo.ru

Abstract. The ever-growing field of microfluidics requires precise and flexible control over fluid flow at

the micro- and nanoscales. Current constraints demand a variety of controllable components for

performing different operations inside closed microchambers and microreactors. In this context, novel

nanophotonic approaches can significantly enhance existing capabilities and provide new functionalities

via finely tuned light-matter interaction mechanisms. Here we propose a novel design, featuring a dual

functionality on-chip: boosted optically-driven particle diffusion and nanoparticle sorting. Our

methodology is based on a specially designed high-index dielectric nanoantenna, which strongly enhances

spin-orbit angular momentum transfer from an incident laser beam to the scattered field. As a result,

exceptionally compact, subwavelength optical nanovortices are formed and drive spiral motion of

peculiar plasmonic nanoparticles via the efficient interplay between curled spin optical forces and

radiation pressure. The nanovortex size is an order of magnitude smaller than that provided by

conventional beam-based approaches. The nanoparticles mediate nano-confined fluid motion enabling

nanomixing without a need of moving bulk elements inside a microchamber. Moreover, precise sorting of

gold nanoparticles, demanded for on-chip separation and filtering, can be achieved by exploiting the non-

trivial dependence of the curled optical forces on the nanoobjects’ size. Altogether, this study introduces a

versatile platform for further miniaturization of moving-part-free, optically driven microfluidic chips for

fast chemical synthesis and analysis, preparation of emulsions, or generation of chemical gradients with

light-controlled navigation of nanoparticles, viruses or biomolecules.

1. Introduction Micro-optofluidics represents one of the most promising and fast growing directions in current

state-of-the-art science and engineering 1–4. In particular, the control of fluid flows in

microsized channels plays an essential role for applications ranging from the transport of

reduced amounts of hazardous or costly substances and DNA biochip technology, to

miniaturized analytical and synthetic chemistry 5–9.

The multidisciplinary nature of microfluidics has brought together seemingly unrelated fields,

such as electrical and mechanical engineering, biology, chemistry and optics. For example, in

the context of chemical engineering, the utilization of distributed microreactors working in

parallel can enhance production significantly and facilitates the design of new products 10,11.

However, slow mixing processes constitute a bottleneck that restricts reaction processes,

especially when the desired reaction rate is high 12,13. For this purpose, fast mixing is highly

required to avoid the reactive process being delayed by this critical step, and to reduce potential

side products 13.

Given the low Reynolds numbers at which fluid flow occurs in microreactors, fluid mixing

represents a significant challenge 14–16. In the most conventional situation where only passive

mixing happens, the main driving mechanism corresponds to diffusion (Brownian motion) 14

implying mixing to take place at a very low rate. Consequently, the effective distance that the

molecules of a fluid need to travel in a mixer before interacting with another fluid with different

composition (i.e. - the mixing length) becomes restrictively long 16. Passive mixers depend

solely on decreasing the mixing length by optimizing the flow channel geometry in order to

facilitate diffusion 14,17. In contrast, active schemes rely on external sources injecting energy into

the flow in order to accelerate mixing and diffusion processes and drastically decrease the

mixing lengths 15,18.

Most early studies related to micromixers have been focused on the passive type. Conversely,

despite their higher cost and complex fabrication methods, the enhanced efficiency of active

micromixers with respect to passive ones has drawn the attention of the scientific community in

the recent years 15. Because of the power and size constraints involved in microfluidics, research

efforts have been focused on the utilization of mixing principles not involving moving

mechanical parts such as surface tension-driven flows 19, ultrasound and acoustically induced

vibrations 20,21, and electro- and magneto-hydrodynamic action 16,22.

Given the small operation scales of microfluidics, micron-scale focusing of laser beams, as

well as different types of light matter interactions make possible to provide sufficiently strong

optical forces to propel particles 23–25, sort objects according to their size or optical properties 26–28 or self-arrange colloidal particles into optically bound structures 23–25,29,30 . Nowadays the

additional degrees of freedom offered by complex shaping of laser beams 31–35 have made

possible the manipulation and trapping of large amounts of microparticles36. In particular, they

allow to create optical vortices with helical phase front (e.g. Laguerre-Gaussian or higher-order

Bessel-beams) carrying both linear and angular momentum37,38. When such an optical micro-

vortex is scattered by particles, it induces an optical torque on them leading to their orbital

motion around the focus of the laser beam39. Due to the angular momentum conservation,

elastic scattering of a circularly polarized beam possessing spin-angular momentum40, by

optically anisotropic 41–43 or non-spherical objects44,45 leads to their spinning oriented along the

direction of propagation of the incident illumination. Combining both types of optical angular

momentum leads to complex spin-orbital interaction 38,46 and novel interesting phenomenon,

e.g. detection of spin forces47,48.

At the nanoscale, metal-based plasmonics dominates and provides exciting means for trapping

and manipulating nanoobjects 49–52. On the other hand, the recently growing field of all-

dielectric nanophotonics53 presents itself as a promising alternative for the integration of

optomechanical concepts in microfluidic devices. Properly designed dielectric nanostructures

with finely tuned Mie resonant response provide the means for tailoring electric and magnetic

components of the scattered light 53–55. They allow to obtain strong near fields, which induce

substantial optical forces acting upon other subwavelength scatterers dispersed in the medium

surrounding the nanostructure 56–60.

In this work, we focus on the conversion of spin angular momentum (SAM) of an incident

circularly polarized plane wave into orbital angular momentum (OAM61,62) of the scattered field

mediated by a specially designed nanostructure constituted of a realistic high refractive index

material (silicon). In contrast to the above-mentioned methods, the optical vortex field created

in this way is very localized, only reaching a few hundreds of nanometers in diameter.

Moreover, the induced optical forces are strong enough to propel gold (Au) nanoparticles of

particular sizes along spiral trajectories around the nanostructure. Based on the latter effect, we

propose a novel method for mixing fluid in nanovolumes mediated by chemically inert Au

nanoparticles (see Figure 1a). In addition, we take advantage of the size sensitivity of the Au

polarizability to achieve light-mediated nanoparticle separation by means of the same

geometrical configuration (see Figure 1b). The subwavelength size of the investigated optical

nanovortex greatly enhances the length scale of interaction in comparison to the more

conventional approaches involving Bessel beams63,64, opening new directions in light-matter

interaction via light angular momentum exchange. We believe that the proposed simple

geometry for optically driven diffusion boosting and nanoparticle sorting is of high interest for a

plethora of applications in microfluidics and lab-on-a-chip devices.

Figure 1. An artistic view of the proposed active nanomixing scheme (left) and radial separation of

nanoparticles (right). (a) A silicon nanocube submerged in a water solution is illuminated by a

circularly polarized laser beam coming from the top. The scattered field carries a nonzero tangential

component of the pointing vector in the xy plane, which induces nonzero orbital angular momentum

in the negative z direction. The same effect causes the spiral motion of Au nanoparticles around the

nanocube. Viscous friction between the moving nanoparticles and the fluid gives rise to convective

fluid motion and enhances fluid mixing. (b) Sorting concept. Nanoparticles of different sizes having

opposite signs of the real part of polarizability are radially displaced in opposite directions – the

smaller ones move towards the nanocube while larger ones move away from it.

2. Formation of an optical nanovortex

The spiral motion of nanoobjects in an optical nanovortex driven by an out-of-plane light source

(Fig. 1a) requires, on the one hand, efficient transformation of spin angular momentum (SAM)

of light to in-plane orbital angular momentum (OAM) of the highly confined scattered near

fields of the nanocube, which, in turn, should be transferred to the nanoparticles. Therefore, as a

first constraint, sufficient in-plane scattering from the nanocube should take place. This urgent

functionality could be enabled, in particular, by the recently observed Transverse Kerker Effect 65 allowing for lateral-scattering only. On the other hand, azimuthal forces arising due to helicity

inhomogeneities in the scattered near field (curl spin forces66) also enable rotational motion.

Hereinafter, we optimize both effects taking into account that we, actually, do not require the

total suppression of forward and backward scattering as in 65, and, therefore, we can tune the

parameters in order to obtain an enhanced optical subwavelength vortex.

A cubical Si nanostructure with refractive index 4n (e.g., silicon at the visible range) and

edge length 250 nm is illuminated by a circularly polarized plane wave propagating along the

negative z -axis (see inset of Figure 2a). For such an incident field one can calculate the angular

momentum flux density using the expressions for paraxial waves 67. The z component can then

be written in the general form

*0(2

2)z inc zn s

z

i cJc

c r E LE L

, (1)

where incE is the electric field, 0 is the vacuum permittivity, L is the orbital angular

momentum operator68, and 2 *0 / 4ms n i L E E is the electric contribution to the SAM flux

density69 of the beam in a medium with refractive index mn . The first term in the right-hand

side of (1) corresponds to the OAM carried by incE . Substituting the expression of a circularly

polarized plane wave into (1) yields the OAM term equal to zero and the total angular

momentum flux density is entirely given by the SAM flux density:

0zJ

I

, (2)

where the wave helicity takes the values of +1 and -1 for left and right-circular

polarization, respectively, and 0I is the incident light intensity. Since the nanostructure has

negligible losses, the total angular momentum of incident and scattered light is conserved. This

conservation law for the total angular momentum implies that the part of the incident SAM,

given by Eq. (2), is transferred to both SAM and OAM of the scattered field.

We can write the total angular momentum surface density of the scattered wave in full

analogy with classical mechanics 68 as

s

c

r SJ , (3)

where sS denotes the time-averaged scattered Poynting vector. Since zJ is non-zero due

to Eq.(2), Eq. (3) implies that the Poynting vector of the scattered field has non-zero tangential

components.

The transverse components of the Poynting vector in the field scattered by the nanocube

display similar rotating features as for chiral scatterers such as helices or gammadion-like

structures 70–72, however the fabrication process of an isotropic nanocube is much less complex

and no chirality is involved. In order to gain better physical insight, we consider the multipole

decomposition for the scattering cross-section of the nanoparticle illuminated with LCP light

deposited over a glass substrate (Figure 2a) 73, and find the most pronounced vortex-like energy

flux at a close vicinity to transverse Kerker point ( 788 nm ) – at the magnetic quadrupole

(MQ) resonance (Figure 2b, 765 nm ). Figure 2a presents the result in the form of the

spectral dependence of the scattering cross-sections of individual multipoles and the total cross-

sections.

In particular, the magnetic quadrupole (MQ) mode presents a very high signal-to-noise-ratio

with respect to the other leading multipoles; the magnetic dipole and electric quadrupole are

almost one order of magnitude smaller and are both out of resonance, while the electric dipole

radiation is suppressed by an anapole state coinciding with the MQ resonant frequency. Thus,

“pure” MQ fields can be obtained, which provide stronger near-field effects in comparison with

lower quality resonances. The resonant MQ mode displaying negligible contributions of the

other multipoles enhances the vorticity of the Poynting vector74 (Figure 2b).

Figure 2. (a): Cartesian multipole decomposition of the scattering cross-section of the dielectric cube

deposited on a glass substrate and centered at the origin of the coordinate system; the cube is illuminated

by a left-hand circularly polarized plane wave propagating against the z-axis. The geometry is illustrated

in the top inset and the ambient medium is air. The dashed black line indicates the position of the resonant

MQ mode. The dashed grey curve corresponds to the total scattering cross section for linearly polarized

incident wave. The total scattered power with incident LCP illumination is obtained from the sum of

contributions of individual multipoles (Total LCP). (b) Colorplot denotes the norm of the total electric

field at the resonant MQ wavelength in the transverse x-y plane at z=100 nm. The arrows indicate

direction and their lengths illustrate the relative size of the transverse part of the Poynting vector.

While the numerical results shown in Figure 2 provide a clear link between the enhanced

transfer of incident field SAM to scattered field OAM at the MQ resonance, a complete physical

picture requires a deeper theoretical insight on the behavior of the fields produced by the MQ

mode under the prescribed illumination. For that purpose, we now express the amount of power

extracted from the field by the nanocube (referred to as the extinction power) 68 as

* 31Re ( )

2 pext incV

P d E j r r , (4)

where the integration is carried inside the volume of the nanocube pV of the nanocube, and ( )j r

is the induced current. We consider incE to be a left-hand circularly polarized plane wave with

the form 0( )inc x yE z i E r e e , with 00 0

ik zE z E e

. Performing the multipole expansion of

( )j r in Eq. (4)75 in the Cartesian basis and substituting the expression of incE , we obtain the

extinction power of the MQ mode

2

* *0 Re (0 )4

) (0ext zx y zy xPk

E MM E , (5)

where ijM is ij -th component of the magnetic quadrupole tensor and ,x yE E indicate the

components of incE , respectively. The center of the nanocube is placed at the origin of the

coordinate system, with the x , y and -axisz oriented perpendicular to its sides. Due to its

inherent rotational symmetry, the optical response of the nanocube is identical for plane waves

linearly polarized in the x or -axisy , which gives zy zxM M , in full agreement with the

numerical results presented in Figure 2a. Therefore, under this condition, and neglecting

reflection from the glass substrate, Eq. (5) can be simplified as

2

0 0 Im2

ext zx

E kP M . (6)

Several conclusions can be readily drawn from Eqs. (5) and (6). Firstly, due to the chosen

geometry, only the zxM and zyM components of the magnetic quadrupole tensor are excited

(since we work in the irreducible Cartesian multipole representation, the multipole tensors are

symmetric and therefore the xzM and yzM components are also excited). Secondly, due to the

symmetry of the system, the contributions of zxM and zyM components to the extinction over

an oscillation period are equal. Therefore, the total extinction power at the MQ resonance

excited by the circularly polarized incident wave is exactly two times larger comparing to a

linearly polarized incident wave. Since we assume negligible absorption, the conclusions made

for extinction are also valid for scattering cross sections shown in Figure 2a. The intuitive

physical picture is the following: during an oscillation period, the incident circularly polarized

electric field gradually changes its polarization between the x and y-axis. Consequently, the

components of the excited magnetic quadrupole tensor oscillate accordingly. The scattered

electric field receives contributions from two magnetic quadrupole moments / 2 delayed from

each other. In analogy with a rotating electric dipole 76, the scattered near-field at the MQ

resonance can be obtained as a superposition of the fields generated by zyM and a / 2

delayed zxM component with equal amplitudes. Due to the phase delay, the scattered Poynting

vector acquires a nonzero tangential component sS

77:

2 2 4 420 0 2

2 70 0

9 33cos sin

16

zxsr k r kM

Sk rc

, (7)

where is the polar angle in spherical coordinates, and r r . In the x-y plane / 2

and 0sS . The total Poynting vector, however, also includes an interference term between the

scattered electromagnetic field and the incident one, which leads to non-negligible curl in the x-

y plane. This is indeed what is observed in the numerical simulations (see Figure 3a, where Pz

is proportional to S ). Substituting the scattered Poynting vector in Eq. (3), the time-averaged

scattered angular momentum density component in the z-axis zJ can be determined as

sz

rJ S

c . (8)

Equation (8) provides direct evidence that SAM from the incident wave has been transferred

to the scattered field giving rise to the optical vortex shown in Figure 2b. Moreover, since zJ

depends on the choice of origin of the coordinate system 78 it can be directly correlated with the

extrinsic OAM of the scattered field. Further inspection of Eqs. (7) and (8) also shows that the

tangential component of the Poynting vector as well as the angular momentum scale

quadratically with the amplitude of the MQ moment, enhancing the field vorticity at the MQ

resonance. Since the angular momentum scales as nr (where n is a positive integer) in the

near field, the vorticity of the Poynting vector is very high close to the particle, but decreases

very fast going away from it, as confirmed in Figure 2b. The latter has very important

consequences regarding the optical forces governing the motion of plasmonic nanoparticles

under the influence of such a field, as we study below.

In this section, we have provided analytical expressions describing the excitation of the MQ

mode with circularly polarized incident light and proposed an intuitive physical picture

explaining the multipolar origin of the tangential component of the Poynting vector giving rise

to an optical vortex in the near field of the nanocube.

3. Optical nanovortex-mediated forces and torques

We can now proceed to study the effect of the scattered field on small dipolar particles. The

time-averaged optical force oF acting upon an induced electric dipole (a nanoparticle),

illuminated with the optical nanovortex field can be written as 69

"

* 0

20

1

2

mo s

ckR

ne

c

F LE E S

, (9)

where mn is the refractive index of the host medium, E is the sum of the incident and

scattered electric fields, and and are the real and imaginary parts of the particle dipole

polarizability, respectively. The first term on the right-hand side of Eq. (9) corresponds to

conservative (curl-free) gradient optical forces acting upon the nanoparticle, which for positive

drags the nanoparticle towards the region of maximal field intensity. The terms in round

brackets describe non-conservative or “scattering” optical forces, hereinafter noted as scF .

The latter receives contributions from the total Poynting vector S and the electric field

contribution to the SAM flux density sL 69. A MQ mode corresponds to an object of well-

defined parity, i.e. a transverse electric (TE) multipole. At the resonance, pure electric or

magnetic multipoles strongly break electromagnetic duality, and, consequently, do not present a

well-defined helicity74,79. This effect manifests itself strongly in the near field66, and implies that

the SAM flux density, which is linked to the helicity density74, features a non-uniform spatial

distribution. Therefore, the second term in scF , acknowledged as the curl spin force66, is not

only non-negligible but also contributes significantly to the total force and torque exerted on

dipolar particles. Employing Eqs. (3) and (9), the z component of the optical torque z acting

upon the nanoparticle due to scF is found to be proportional to the tangential components of

the Poynting vector S and the curl of sL :

"

0

20

1z sc sz

mr Sck n

c

r F L

, (10)

with r the distance to the z-axis. Equation (10) clearly illustrates that the amount of orbital

torque transmitted to the particles depends on the radiation pressure and the helicity spatial

distribution of the scattered field by means of S and sL , as well as the optical response of

the particle itself by means of " . Interestingly, in the case of the MQ, it is straightforward to

show that the contribution of the scattered field to the curl spin force only has an azimuthal

component, i.e. it only induces orbital motion. This result is general to any magnetic (TE)

multipole field. The interference with the incident illumination leads, however, to important

radial and polar components (see Figure 3).

Currently, very few groups66,80 have investigated optical fields where the effect of spin curl

forces can be visibly appreciated in the dynamics of moving nanoparticles in a fluid. In contrast,

our calculations directly prove that both spin and radiation pressure contribute to the induced

optical torque in the scattered near field of the dielectric cube.

In Figure 3, we show the optical torque (Figure 3a) experienced in the near field by an

arbitrary absorbing 40 nm radius spherical nanoparticle with 2n i calculated with Eq.(10) and

averaged over several circular rings on parallel transverse planes (perpendicular to the incident

propagation direction). Remarkably, particles whose centers of mass are located at different

heights experience different contributions from the curl-spin ( Spinz ) and radiation pressure ( P

z )

torques, as can be visually appreciated in the force field plots shown in Figure 3b. Interestingly,

the direction of the curl-spin torque can be opposite to the helicity of the incident wave,

contrarily to radiation pressure.

In our setup, the particles are initially pushed towards the glass substrate by the incident

beam intensity, where they experience a combination of radiation pressure and curl spin torques

(Figure 3b (II)). It is worth noting that Pz is nonzero at z=0, contrarily to what one might

initially expect from Eq.(7), but we once more emphasize that the total Poynting vector entering

in Eq.(10) includes an additional interference term between the incident and scattered field

yielding a small azimuthal component.

Figure 3. (a) Optical torque affecting absorbing 40 nm radius nanoparticles with 0.5 2n i in the near

field of the MQ resonance ( 765 nm ). The time-averaged spin (Spinz ) and radiation pressure (

Pz )

contributions have been spatially averaged ( ) in parallel x-y planes in the near field along the height

of the cube (z axis). (b) Transverse cuts at z=50 nm (I) and z=-120 nm (II) showing the vector field

distributions of the x and y components of the total and curl-spin forces around the cube.

To summarize, we have introduced analytical expressions for the optical forces and torque

induced on small dipolar absorbing particles, which allowed us to unambiguously distinguish

the contributions of the scattering and the spin forces. The numerical calculations presented in

Figure 3 demonstrate that both effects mediate the strongly confined (subwavelength) particle

rotation with respect to the z-axis (i.e. the direction of propagation of the incident wave).

4. Nanoparticle dynamics in the optical nanovortex

We now turn our attention towards the potential applicability of the considered effect as a

mixing method for microfluidic reactors. In order to illustrate the concept, the high-index cube

is placed in a water host medium ( 1.335mn ), containing chemically inert, biologically

compatible nanoparticles. The dynamics of the latter will be affected by the optical forces

arising due to the interaction with the cube’s scattered field together with the Brownian and

viscous drag forces induced in the fluid. The obvious and most convenient candidates to act as

mixing mediators are gold (Au) nanoparticles, because they would not interact with the

chemical and/or biological compounds dissolved in the solutions and are utilized in a broad

range of microfluidics applications81,82

In order to increase the mechanical orbital torque transferred to the Au nanoparticles and to

prevent them from sticking to the walls of the nanocube due to attractive gradient forces, the

ratio /sc oF F should be maximized. For high enough ratios, non-conservative scattering

forces govern the nanoparticle dynamics, causing them to undergo spiral paths around the

nanocube and act as stirrers, enhancing convective fluid motion and thus diffusive mixing of

any admixtures present in the water solution.

Under the influence of a given optical field, the scattering force can be the leading force

acting upon the nanoparticle only if the real part of the nanoparticle polarizability is negligible

in contrast to the imaginary one (see Eq.(9)). For simplicity, we assume a spherical shape so that

their dipole polarizability can be evaluated analytically with the exact Mie theory formulae by

the method described in Refs.83–85:

013

6, d p

dd p

d

k R ik

k Ra

, (11)

where dk and d are the wavenumber and relative permittivity of water, respectively. 1a

denotes the first order electric Mie coefficient 86, which depends on the refractive index contrast

between the particle and the medium and the dimensionless parameter d pk R , where pR is the

nanoparticle radius. Figure 4 shows the real and imaginary parts of the polarizability for Au

particles of different sizes dispersed in water. The calculations are performed for the well-

known optical dispersion properties of bulk Au 87 ( Aub ), taking into account the Drude size

corrections due to the limitation of the electron mean free path in small metallic particles 88

2 2

'

' 0.7

pl p

p

lAu Aup b

b b

b bF

i

v

R

i

, (12)

where pl , b and Fv are the plasma resonant frequency, the damping constant from the free

electron Drude model, and the Fermi velocity, respectively.

Figure 4. Real and imaginary parts of the dipole polarizability are calculated from Eqs. (11) and (12) for

Au nanoparticles of different sizes that are dispersed in water. The excitation wavelengths correspond to

those in free space. For nanospheres with radius 35 nmpR , the real part can be equal to zero while the

imaginary part is enhanced.

Figure 4a demonstrates that in the vicinity of the plasmon resonance nanoparticles with

35 nmpR can fulfill the condition ' 0 with enhanced values of " . For example, for

particles with 40 nmpR , the full suppression of the gradient force occurs at 500 nm and 530

nm (see Figure 4a). Consequently, only scattering forces are allowed for them and the ratio

/sc oF F is maximized.

Large Au nanoparticles with 35 nmpR could be considered to break the limits of the

electric dipole approximation assumed in Eq.(9). To prove its validity for quantitative

calculations of the optical forces, we have compared our results with exact numerical

computations via integrating the Maxwell stress tensor over a 40 nm radius Au nanoparticle and

obtained very good agreement (see Supplementary information). Moreover, the electric field

distribution in the system plotted in Figure S1 (B-D) shows negligible perturbations in the

presence of an Au nanoparticle with no backscattering, further confirming the validity of the

involved approximations.

In order to determine the trajectories of the Au nanoparticles in water, let us consider the

radiation pressure coming from the incident beam along the z axis to be completely

compensated in the presence of the glass substrate. Therefore, the trajectories can be treated as

two-dimensional, localized only in the transverse x y plane.

Considering scattering force scF , viscous drag force DF and stochastic Brownian (thermally

activated) forces BF acting on the nanoparticle of mass pm , the equation of motion can be

written in the following form:

B D p psc m F F F r , (13)

where pr is the particle instantaneous acceleration vector. Once the scattering force

distribution is determined, Eq. (13) can be solved in Comsol Multiphysics© software package

utilizing its particle tracing functionality. For small spherical geometries, the Brownian and

viscous forces can be expressed as 89

12 B

B p

k TR

t

F Ψ

, (14)

6D p pR F r , (15)

where is the dynamic viscosity of water ( 48.9 10 Pa s at ambient temperature), Ψ is a

dimensionless vector function of randomly distributed numbers with zero mean 89, T is the

temperature of the system and Bk is Boltzmann’s constant. The numerical solver models the

Brownian forces as a white noise random process with a fixed spectral intensity implying the

amplitudes of the force to depend on the iterative time step t 89. Formula (15) corresponds to

Stokes’ law. Its validity is only justified for very low Reynolds numbers 90 being, actually, the

case for a microfluidic chip 15. We assume the system to be at ambient temperature.

Figure 5. Trajectories of Au nanoparticles of 40 nm radius during 1ms of simulations. Illumination

wavelength in vacuum was 530 nm wavelength (in water 396 nm). (a) – No incident illumination, only

Brownian motion and drag forces act on the particles; (b) – The nanocube is illuminated with a circularly

polarized light and the optical force significantly contributes. The Au nanoparticles spirally move around

the cube. The figures are scaled to the length of the cube side equal to 158 nm.

The parameters for the simulations are given in Tables S1, S2 of the Supplementary

Information. We consider that Au nanoparticles of 40 nm radius are uniformly distributed

around the Si nanocube. Their trajectories during a simulation time of 0.1 ms are shown in

Figure 5. If the nanocube is not illuminated (Figure 5a), thermal activation induces random

movements of the nanoparticles independently on their position in the simulation domain.

Conversely, when the system is illuminated with circularly polarized light with intensities in the

order of 280 m50 W/μm , (corresponding to typical values utilized in conventional optical

trapping schemes91), a sufficient mechanical torque is transferred to the Au nanoparticles and

drives them along spiral trajectories (Figure 5b).

Equation (7) reveals that S becomes negligible far from the nanocube. Furthermore, the

numerical simulations show that the curl spin force has no longer a significant effect.

Consequently, Brownian motion and conventional radiation pressure dominate the dynamics

(see Figure 6b). It is thus natural to introduce the effective radius mr which specifies the “area

of influence” of the optical nanovortex (red dashed curves in Figure 6). For mr r , the majority

of the Au nanoparticles circulate around the nanocube, and the dielectric nanocube acts as an

effective optical drive for convective stirring of the fluid around it.

Figure 6. (a) Distance from the dielectric nanocube as a function of time for 19 Au nanoparticles initially

evenly distributed along the x axis in the simulation domain depicted in Figure 5b. The particle

trajectories are calculated up to a simulation time of 0.16 ms. Red dashed curves specify the effective

radius mr , where the contribution of optical forces becomes negligible. The optical nanomixing effect is

thus achievable at distances from the nanocube smaller than mr .

The radius mr reaches approximately half of the incident wavelength in water and thus the

mechanical effect of optical vortices upon a nanoparticle is formed in the subwavelength region

and gets stronger closer to the nanocube. Such a reduced scale cannot be reached using any

focused far field, e.g. radial and Bessel beams 63,64,92. Up to our knowledge, this is the first

proposal providing optical nanovortices created in a simple, realizable setup avoiding the need

of lossy plasmonic nanoantennas 93,94, short wavelength guided modes 95 or complex chiral

structures 96. Such optical nanovortices represent a promising component for on-a-chip OAM

exchange driving light-matter interactions (e.g. controlled light emission from quantum dots 96,

super-resolution97,98 and nanoobject manipulation63,92).

5. Nanovortex-mediated liquid mixing

To study in detail the liquid flow driven by the proposed nanomixing design, we once again

utilize direct time-domain simulation in COMSOL Multiphysics©. At each time step, the

particle position and velocity, as well as the fluid pressure and velocity fields are obtained by

solving Eq. (13), the Navier-Stokes, and mass balance equations for the fluid 90. We consider

simplified forms of the last two equations assuming laminar, incompressible flow, in

accordance with the previous results for the particle trajectories. Furthermore, we impose open

boundary conditions at the edges and simulate a large fluid domain around the nanocube

(usually lab-on-a-chip microchambers are of the order of tens of micrometers). An Adapted

Lagrange-Euler method (ALE)99 is implemented in order to accurately interpolate the mesh

displacements induced by the nanoparticle movement.

Figure 7 shows the calculated stresses and velocity fields in the fluid during 200 s . The

particles start with zero initial speed and gradually accelerate under the influence of radiation

pressure and spin forces arising from their interaction with the optical nanovortex. Consequently,

the fluid environment is also displaced, as Figure 7a demonstrates. At longer times, a single-

vortex-like velocity distribution is established as shown in Figure 7c.

Figure 7. Formation of a fluid nanovortex due to the movement of two Au nanoparticles (white

circles moving anticlockwise) driven by optical nanovortex formed around the nanocube (white square).

Background color map denotes the distribution of stress in the fluid in 1/s and white contours show the

fluid velocity field streamlines of the initially static fluid at different times since the nanoparticles became

optically driven. (a) t=0.01 s , (b) t=100 s and (c) t=200 s . The represented domain has dimensions

800 800 nm . Parameters of the simulations are given in Table S1 of the Supplementary information.

Open fluid boundary conditions were imposed at a distance of 1.5 m from the center of the cube. For the

sake of clarity, thermal motion is not taken into account in the simulation.

The velocity streamlines are more inhomogeneous at shorter times, when the nanoparticles start

moving. Already at 100 s , only small fluid distortions take place very close to the

nanoparticles and the nanocube. Therefore, a possible way to further enhance the fluid

nanomixing would be to realize periodic switching between left- and right-hand circularly

polarized incident light, which would reverse the direction of particle motion maintaining a high

level of inhomogeneity in the fluid stress field.

Noteworthy that, while all the previous calculations were performed for Au nanoparticles in

the visible range, similar dynamics can also be obtained for Ag nanoparticles in the UV range of

the spectrum, where ' 0 100. Nanomixing in the UV region could be advantageously

combined with photochemically active processes of the involved chemical compounds.

6. Optical sorting of Au nanoparticles via the nanovortex

Hereinafter, we demonstrate the important capability of the proposed configuration to realize

optical force-mediated particle on-chip sorting. In the following paragraphs, we illustrate a

novel, dynamical, non-contact size sorting method for Au nanoparticles in liquid solutions

addressing one of the most challenging targets of conventional microfluidics with the help of

dielectric nanophotonics.

The proposed method is based on the sign reversal displayed by ' close to the plasmon

resonance as we demonstrated in Figure 4. The transition reverses the direction of the radial

gradient force acting upon the nanoparticle (see the first term in Eq. (9)). At a given incident

wavelength, we can split the behavior of the Au nanoparticles into two regions I and II (see

Figure 8a). Smaller nanoparticles from region I with positive ' are attracted by the radial

gradient force towards the nanocube, while larger nanoparticles, from region II, should be

repelled outwards.

However, in region I, there is a competition between the gradient and scattering forces dragging

the nanoparticles in opposite radial directions. Simulations performed for Au nanoparticles of

radii 20-30 nm proves their movement towards the nanocube (see Figure 8b). For particles

smaller than 20 nm, the nanoparticle polarizabilities are very low and thus the driving optical

forces are negligible in comparison with Brownian forces. Contrarily, nanoparticles with radii

close to or larger than 40 nm (i.e. in the vicinity or inside region II), spiral away from the

dielectric nanocube. The particle dimensions in the latter case might go beyond the dipolar

approximation expected in Eq. (9). However, our numerical simulations prove the correctness of

the previous conclusions (see Supplementary information, Figures S1 and S2).

Figure 8. (a) Real ( ' ), and imaginary ( '' ) parts of the dipole polarizability as a function of the

particle radius pR , for an incident free space wavelength of 530 nm. The black dashed line indicates zero

real or imaginary part, while the red dashed line shows the border between region I and II. (b) Calculated

trajectories for two sets of particles with radii 20 nm (blue – Region I) and 50 nm (red – region II)

demonstrate opposite movement in radial direction.

Figure 8b illustrates the proposed method for nanoparticle separation by comparing the

behavior of two sets of Au nanoparticles with n0 m2pR (blue) and 50 nm (red), respectively.

The first set of smaller particles lies well inside region I, while the second one fits the region II.

As it can be clearly seen, that gradient forces are strong enough to pull smaller nanoparticles

towards the nanocube, tracing an inward curved pattern. Contrarily, strong scattering forces and

repulsive gradient ones acting on larger nanoparticles from region II result in an outward motion.

Therefore, Au nanoparticles spiral around the nanocube inwards or outwards depending on their

size. For the sake of clarity, we have purposely neglected the effect of thermal agitation in

Figure 8b, being aware from the simulations that the latter only produces more intricate

trajectories without affecting the final outcome of the nanoparticles (not shown here).

A precise, in situ size control of Au nanoparticles is a crucial step in many applications

where the processes involved are highly dependent on the latter, e.g. biological cell uptake rates 101,102, toxicity 103 and Raman signal intensity 104.

7. Conclusion

We present conditions for maximal conversion of spin angular momentum of the incident light

to orbital angular momentum of the scattered light via a specially designed transversely

scattering silicon nanocube. The azimuthal component of the Poynting vector of the scattered

field originates from the strong magnetic quadrupole resonance excited in the nanocube. A gold

nanoparticle of appropriate size, illuminated by such optical field and dispersed in the fluid

surrounding the nanocube, experiences a combination of non-conservative spin and radiation

pressure forces with non-zero azimuthal component. They are significant only up to a distance

of about half of the illuminating wavelength from the nanocube. The exceptionally compact

optical nanovortex drives the dynamics of the nanoparticles, inducing a convection fluid flow at

the nanoscale. The direction of particle motion can be reversed simply by flipping the helicity of

the incident circular polarization illuminating the nanocube. The proposed mechanism can serve

as a nanoscale fluid mixer and diffusion booster driven by light in a contact-less and flexible

way. Arrays of the studied dielectric structure can be easily imprinted on the surface of a

microfluidic chip and controllably illuminated in an independent fashion. Hence, we open the

doors to very exciting perspectives such as light controlled mixing or even on-chip directional

fluid navigation. Employing the dependence of the optical properties of gold nanoparticles on

their size, we demonstrate feasibility to drag nanoparticles affected by the optical vortex either

towards or outwards the nanocube. Thus, smaller nanoparticles (20-30 nm in radius) can be

aggregated at the nanocube surface while larger nanoparticles move away from it and drive the

fluid flow. This behavior can be utilized to perform in situ nanoscale size separation directly

inside the microfluidic chip.

The proposed, rather simple concept can be extended to nanostructures of different shapes, to an

array of such nanostructures and to different types of dispersed nanoparticles, e.g. silver

nanoparticles offer an exciting option to combine optical nanovortex with photo-chemistry at

the nanoscale. Our approach opens a new room of opportunities for the integration of simple,

optically driven nanosorting or filtering modules in on-chip platforms paving the way towards

more efficient functionalities in micro- and nano-fluidic systems.

Acknowledgements

ACV, DK, EAG, HKS, DR, SY, and ASS acknowledge financial support from the Russian

Foundation for Basic Research (grants 18-02-00414 and 18-52-00005); the force calculations

were partially supported by Russian Science Foundation (Grant No. 18-72-10127). PZ

acknowledge support of the Technology Agency of the Czech Republic (grant TE01020233)

and the Czech Academy of Sciences.

Conflicts of interest

The authors declare no conflict of interest.

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